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I am solving a question that says if we have sequence $x(n)$ of a signal missing one term then we have to find autocorrelation matrix $R_x$ as follows:

$$R_x = E\{\mathbf {xx^H}\}$$ Now if I take $x(n) = [x(0) , x(2),x(3)]$ that is missing term $x(1)$ I found following value of $R_x$

$R_x = $$\left( \begin{matrix} r_x(0) & r_x^*(2) & r_x^*(3) \\ r_x(2) & r_x(0) & r_x^*(1) \\ r_x(3) & r_x(1) & r_x(0) \\ \end{matrix} \right) $

Now I check that this matrix is Toeplitz or not, by the definition of Toeplitz (constant diagonal term) $R_x$ is Toeplitz. But when I checked the solution it says the above matrix is not Toeplitz and there is no explanation given. I wonder what I am missing in definition of a Toeplitz matrix that does not fit in the above matrix. Can anyone explain this?

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For $\boldsymbol{R}_x$ to be toeplitz it should be:

$$\boldsymbol{R}_x = \begin{bmatrix}r_{x}(0) & r_{x}(2) & r_{x}(3) \\ r_{x}(2) & r_{x}(0) & r_{x}(2) \\ r_{x}(3) & r_{x}(2) & r_{x}(0) \end{bmatrix} $$

Since it is not, it is not a toeplitz matrix.

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  • $\begingroup$ you mean when they say diagonal should be constant means all diagonals should be constants? $\endgroup$
    – Tab
    Commented May 7, 2016 at 3:12
  • $\begingroup$ Basically all the diagonal elements should be the same. If this is what you mean by constant, then yes. $\endgroup$
    – Amal
    Commented May 7, 2016 at 3:16
  • $\begingroup$ oops yes I checked definition again it was just my lack of attention while reading it. $\endgroup$
    – Tab
    Commented May 7, 2016 at 3:16
  • $\begingroup$ I understand. Been there multiple times myself. $\endgroup$
    – Amal
    Commented May 7, 2016 at 3:17

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